Finite Group Databasebeta

← #10849297

Subgroups — #10849297

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic
2C2×C3×#478C211yesmaximal, characteristic
3#83041C211yesmaximal, characteristic, complemented
4C2×#5069C211yesmaximal, characteristic, complemented
5#71609C211yesmaximal, characteristic
6#70642C212yesmaximal, complemented
7#70588C212yesmaximal, complemented
8#70473C211yesmaximal, characteristic, complemented
9#70456C211yesmaximal, characteristic, complemented
10#70216C212yesmaximal
11#70216C212yesmaximal
12#25656—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#936C2212yes—
16C2×#926—21no—
17C2×#924—21no—
18#5542—24no—
19#5542—24no—
20C2×#887C2212yes—
21C2×#879—22no—
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#478—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×#832C2212yes—
31C2×#816C2212yes—
32C2×#816C2212yes—
33#5069—24no—
34#4981—21no—
35#4979—21no—
36#4976—22no—
37#4972C2212yes—
38#4972C2212yes—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C4×C4⋊Dic3C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#491—31no—
48C2×#478D311yescharacteristic
49#2584—31no—
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1981—31no—
54#1865—32no—
55#1863—32no—
56#1762—31no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×Dic6D411yescharacteristic
60C22×Dic6C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#125—22no—
64C3×#125—22no—
65C3×C4×D4—22no—
66C3×C4×D4D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76#936—24no—
77#936—24no—
78#926—28no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3C2312yes—
83#924—28no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4D412yes—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2312yes—
92#887—24no—
93#887—24no—
94#879—216no—
95#869—22no—
96#869D414yes—
97C4×Dic6—22no—
98C4×Dic6—22no—
99C3×#107—22no—
100C3×#106—22no—
101C3×C4⋊C8—22no—
102C3×#102—22no—
103C3×#102—22no—
104#832—24no—
105C3⋊#95—21no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108#816—24no—
109#816—24no—
110#810—24no—
111#810—24no—
112#803—22no—
113#556—64no—
114#556—64no—
115C2×C4⋊Q8—31no—
116C2×#126—62no—
117C2×#125C2×D311yescharacteristic
118C2×#124—32no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×D4C2×D311yescharacteristic
123C2×#117—61no—
124#491—64no—
125#478—24no—
126C2×#107C2×D311yescharacteristic
127C2×#106C2×D311yescharacteristic
128C2×C4⋊C8C2×D311yescharacteristic
129C2×#103—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#102—32no—
134C2×#102C2×D311yescharacteristic
135C2×#102C2×D311yescharacteristic
136C2×#97—31no—
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4—31no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—21no—
152C22×C3×C4—24no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—44no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D412yes—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—42no—
170C2×Dic6C2×D414yes—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D412yes—
183C3×C22⋊C4—24no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186#217—28no—
187#217—28no—
188C4⋊Dic3—22no—
189C4⋊Dic3—24no—
190C4⋊Dic3—24no—
191C4⋊Dic3—22no—
192C4⋊Dic3#4114yes—
193Dic3⋊C4—28no—
194Dic3⋊C4—24no—
195Dic3⋊C4—44no—
196Dic3⋊C4—82no—
197C4×Dic3—28no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—81no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—31no—
205C22×Q8—32no—
206C22×D4C22×D311yescharacteristic
207C23×C4—21no—
208C22×C8—32no—
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211C4⋊Q8—34no—
212C4⋊Q8—62no—
213#126—616no—
214#125—22no—
215#125—22no—
216#124—64no—
217#124—64no—
218C22⋊Q8—68no—
219C4×Q8—64no—
220C4×Q8—64no—
221C4×Q8—62no—
222C4×Q8—62no—
223C4×D4—22no—
224C4×D4#6714yes—
225#117—68no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—62no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C42—32no—
243C2×C42—62no—
244C2×C42—61no—
245C2×C42C22×D311yescharacteristic
246#107—22no—
247#106—22no—
248C4⋊C8—22no—
249#103—64no—
250#103—64no—
251#103—64no—
252#103—64no—
253#102—22no—
254#102—22no—
255#102—64no—
256#97—62no—
257#95—62no—
258#95—62no—
259#95—61no—
260C23×C3—22no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3C22×D411yescharacteristic
266C3×D4—22no—
267C3×D4—22no—
268C3×D4—42no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—48no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4#12012yes—
281C2×C3×C4#12012yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×D411yescharacteristic
284C2×C3×C4C2×#4112yes—
285C2×C3×C42−1+412yes—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298Dic6—24no—
299Dic6—24no—
300Dic6—44no—
301Dic6—42no—
302Dic6—42no—
303C3×C8—41no—
304C3×C8—41no—
305C3⋊C8—42no—
306C24—21no—
307C2×Q8—34no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—122no—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4C2×#6712yes—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—32no—
324C22×C4—32no—
325C22×C4—44no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8—62no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—34no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—68no—
347C4⋊C4—62no—
348C4⋊C4—64no—
349C4⋊C4—64no—
350C4⋊C4—64no—
351C4⋊C4—62no—
352C4⋊C4—124no—
353C4⋊C4—242no—
354C4⋊C4C2×#6712yes—
355C22⋊C4—24no—
356C22⋊C4—68no—
357C22⋊C4—68no—
358C42—68no—
359C42—62no—
360C42—62no—
361C42—241no—
362C42D3×D412yes—
363C42C2×#6712yes—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3C2×#12011yescharacteristic
370C22×C3#54812yes—
371C22×C3#57512yes—
372C22×C3#57711yescharacteristic
373C22×C3#57811yescharacteristic
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—81no—
378C3×C4—81no—
379C3×C4#53211yescharacteristic
380C3×C4#53611yescharacteristic
381C3×C4#54712yes—
382Dic3—44no—
383Dic3—44no—
384Dic3—42no—
385Dic3—42no—
386C23—22no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23C2×D3×D411yescharacteristic
392Q8—64no—
393Q8—64no—
394Q8—124no—
395Q8—122no—
396Q8—122no—
397D4—22no—
398D4—22no—
399D4—42no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—64no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—64no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4#93712yes—
424C2×C4#93712yes—
425C2×C4C2×D3×D411yescharacteristic
426C2×C4#100712yes—
427C2×C4D3×#4112yes—
428C2×C4C22×#6711yescharacteristic
429C8—41no—
430C8—41no—
431C8—122no—
432C2×C3—82no—
433C2×C3#243211yescharacteristic
434C2×C3#300511yescharacteristic
435C2×C3#301311yescharacteristic
436C2×C3#337612yes—
437C2×C3#337712yes—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22#563312yes—
444C22#577011yescharacteristic
445C22#577111yescharacteristic
446C22#577312yes—
447C22C2×#93711yescharacteristic
448C4—22no—
449C4—22no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—124no—
454C4—124no—
455C4—122no—
456C4—122no—
457C4#563112yes—
458C4#581611yescharacteristic
459C4#583211yescharacteristic
460C3#2565611yescharacteristic, complemented, Sylow
461C2—82no—
462C2#8115412yes—
463C2#8115512yes—
464C2#8275911yescharacteristic
465C2#8391311yescharacteristic
466C2#8394511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#478C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#478D3
hypocenterC3#25656
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #83041C2semidirect
4. C2×#5069C2semidirect
6. #70642C2semidirect
7. #70588C2semidirect
8. #70473C2semidirect
9. #70456C2semidirect
460. C3#25656semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55422
C2×C3×#4782
#830412
C2×#50692
#716092
#706422
#705882
#704732
#704562
#702162
#256563